Operator-based Comparative Study of Anomalous Thermal Diffusion

For decades, anomalous diffusion has attracted growing attention as a fundamental transport mechanism in complex and heterogeneous systems. In media such as porous materials, polymers, and biological tissues, transport often deviates from classical diffusion, so conventional diffusion models fail to describe these phenomena. Memory effects, nonlocal interactions, structural heterogeneity, and strong correlations between constituents typically cause these deviations. Fractional diffusion equations have emerged as powerful tools to overcome these limitations, extending classical approaches by incorporating memory kernels and nonlocality. In this study, we investigate and compare solutions of fractional diffusion equations formulated using the Caputo, Caputo–Fabrizi, and Atangana–Baleanu operators. Analytical solutions are derived and systematically analyzed to investigate the specific impact of different fractional operators on diffusion profiles, transport dynamics, and memory-dependent behavior, highlighting how the choice of operator influences these characteristics. Our comparative analysis reveals notable differences in transport patterns across operators, offering further insights into the interplay among memory effects, structural heterogeneity, and boundary-induced effects. This study provides a comprehensive framework for modeling subdiffusive transport and demonstrates the effectiveness of different fractional operators in capturing anomalous diffusion phenomena in complex media. The findings have potential applications in photothermal characterization and imaging of various functional materials and biological tissues.